Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.1% → 97.0%

Time: 22.2s

Alternatives: 11

Speedup: 28.1×

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \frac{\cos k}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+105)
   (* (/ (/ l (tan k)) k) (/ (/ 2.0 (/ (sin k) l)) (* k t)))
   (* (/ 2.0 (pow (* (sin k) (/ k l)) 2.0)) (/ (cos k) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+105) {
		tmp = ((l / tan(k)) / k) * ((2.0 / (sin(k) / l)) / (k * t));
	} else {
		tmp = (2.0 / pow((sin(k) * (k / l)), 2.0)) * (cos(k) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d+105) then
        tmp = ((l / tan(k)) / k) * ((2.0d0 / (sin(k) / l)) / (k * t))
    else
        tmp = (2.0d0 / ((sin(k) * (k / l)) ** 2.0d0)) * (cos(k) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+105) {
		tmp = ((l / Math.tan(k)) / k) * ((2.0 / (Math.sin(k) / l)) / (k * t));
	} else {
		tmp = (2.0 / Math.pow((Math.sin(k) * (k / l)), 2.0)) * (Math.cos(k) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e+105:
		tmp = ((l / math.tan(k)) / k) * ((2.0 / (math.sin(k) / l)) / (k * t))
	else:
		tmp = (2.0 / math.pow((math.sin(k) * (k / l)), 2.0)) * (math.cos(k) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+105)
		tmp = Float64(Float64(Float64(l / tan(k)) / k) * Float64(Float64(2.0 / Float64(sin(k) / l)) / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 / (Float64(sin(k) * Float64(k / l)) ^ 2.0)) * Float64(cos(k) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e+105)
		tmp = ((l / tan(k)) / k) * ((2.0 / (sin(k) / l)) / (k * t));
	else
		tmp = (2.0 / ((sin(k) * (k / l)) ^ 2.0)) * (cos(k) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+105], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000046e105

   1. Initial program 33.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 33.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 33.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 34.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 34.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 35.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 35.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 46.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 46.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 46.6%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 54.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 89.4%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 89.4%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 89.4%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 95.1%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 95.1%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 89.4%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 89.4%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 89.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 89.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 92.2%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 92.2%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 92.2%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 92.2%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 92.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 97.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 97.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 97.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 97.9%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 95.1%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   12. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   13. Step-by-step derivation

       1. times-frac 99.2%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot t}} \]

       2. associate-/l* 99.2%

          \[\leadsto \frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\color{blue}{\frac{2}{\frac{\sin k}{\ell}}}}{k \cdot t} \]

   14. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}} \]

   if 5.00000000000000046e105 < (*.f64 l l)

   1. Initial program 24.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. +-rgt-identity 13.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]

      2. associate-*l* 13.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]

      3. mul0-rgt 17.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]

      4. distribute-lft-in 21.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]

      5. +-rgt-identity 24.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      6. sub-neg 24.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}\right)} \]

      7. +-commutative 24.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + \left(-1\right)\right)\right)} \]

      8. associate-+l+ 25.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}\right)} \]

      9. metadata-eval 25.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \color{blue}{-1}\right)\right)\right)} \]

      10. metadata-eval 25.7%

          \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

      11. +-rgt-identity 25.7%

          \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   3. Simplified 25.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 23.1%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

      2. expm1-udef 11.0%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} - 1}} \]

   5. Applied egg-rr 17.5%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left({\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}\right)} - 1}} \]
   6. Step-by-step derivation

      1. expm1-def 18.4%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}\right)\right)}} \]

      2. expm1-log1p 18.6%

         \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)}^{2}}} \]

      3. *-commutative 18.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}}^{2}} \]

      4. *-commutative 18.6%

         \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)}\right)}^{2}} \]

   7. Simplified 18.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

   8. Taylor expanded in k around inf 39.9%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. associate-/l* 39.9%

         \[\leadsto \frac{2}{{\left(\color{blue}{\frac{k}{\frac{\ell}{\sin k}}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

      2. associate-/r/ 40.0%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\frac{k}{\ell} \cdot \sin k\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

   10. Simplified 40.0%

       \[\leadsto \frac{2}{{\color{blue}{\left(\left(\frac{k}{\ell} \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   11. Step-by-step derivation

       1. div-inv 40.0%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\left(\frac{k}{\ell} \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]

       2. *-commutative 40.0%

          \[\leadsto 2 \cdot \frac{1}{{\left(\color{blue}{\left(\sin k \cdot \frac{k}{\ell}\right)} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

   12. Applied egg-rr 40.0%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 40.0%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}} \]

       2. metadata-eval 40.0%

          \[\leadsto \frac{\color{blue}{2}}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}} \]

       3. unpow2 40.0%

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right) \cdot \left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}} \]

       4. swap-sqr 39.1%

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sqrt{\frac{t}{\cos k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}} \]

       5. unpow2 39.1%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \left(\sqrt{\frac{t}{\cos k}} \cdot \sqrt{\frac{t}{\cos k}}\right)} \]

       6. rem-square-sqrt 98.0%

          \[\leadsto \frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \color{blue}{\frac{t}{\cos k}}} \]

   14. Simplified 98.0%

       \[\leadsto \color{blue}{\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]
   15. Step-by-step derivation

       1. expm1-log1p-u 52.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\right)\right)} \]

       2. expm1-udef 31.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\right)} - 1} \]

   16. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 52.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\right)\right)} \]

       2. expm1-log1p 98.0%

          \[\leadsto \color{blue}{\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}} \]

       3. associate-*r/ 98.0%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2} \cdot t}{\cos k}}} \]

       4. associate-/l* 98.0%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}{\frac{\cos k}{t}}}} \]

       5. associate-/r/ 98.6%

          \[\leadsto \color{blue}{\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \frac{\cos k}{t}} \]

   18. Simplified 98.6%

       \[\leadsto \color{blue}{\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \frac{\cos k}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \frac{\cos k}{t}\\ \end{array} \]

Alternative 2: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+259}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot t_1}{\sin k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))))
   (if (<= (* l l) 0.0)
     (* (/ l k) (/ (/ (* l 2.0) (sin k)) (* k (* k t))))
     (if (<= (* l l) 1e+259)
       (* (/ 2.0 k) (/ (/ (* l t_1) (sin k)) (* k t)))
       (* t_1 (* 2.0 (/ (/ l (* k k)) (* (sin k) t))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / k) * (((l * 2.0) / sin(k)) / (k * (k * t)));
	} else if ((l * l) <= 1e+259) {
		tmp = (2.0 / k) * (((l * t_1) / sin(k)) / (k * t));
	} else {
		tmp = t_1 * (2.0 * ((l / (k * k)) / (sin(k) * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k)
    if ((l * l) <= 0.0d0) then
        tmp = (l / k) * (((l * 2.0d0) / sin(k)) / (k * (k * t)))
    else if ((l * l) <= 1d+259) then
        tmp = (2.0d0 / k) * (((l * t_1) / sin(k)) / (k * t))
    else
        tmp = t_1 * (2.0d0 * ((l / (k * k)) / (sin(k) * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / k) * (((l * 2.0) / Math.sin(k)) / (k * (k * t)));
	} else if ((l * l) <= 1e+259) {
		tmp = (2.0 / k) * (((l * t_1) / Math.sin(k)) / (k * t));
	} else {
		tmp = t_1 * (2.0 * ((l / (k * k)) / (Math.sin(k) * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.tan(k)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (l / k) * (((l * 2.0) / math.sin(k)) / (k * (k * t)))
	elif (l * l) <= 1e+259:
		tmp = (2.0 / k) * (((l * t_1) / math.sin(k)) / (k * t))
	else:
		tmp = t_1 * (2.0 * ((l / (k * k)) / (math.sin(k) * t)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(l / k) * Float64(Float64(Float64(l * 2.0) / sin(k)) / Float64(k * Float64(k * t))));
	elseif (Float64(l * l) <= 1e+259)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * t_1) / sin(k)) / Float64(k * t)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(l / Float64(k * k)) / Float64(sin(k) * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (l / k) * (((l * 2.0) / sin(k)) / (k * (k * t)));
	elseif ((l * l) <= 1e+259)
		tmp = (2.0 / k) * (((l * t_1) / sin(k)) / (k * t));
	else
		tmp = t_1 * (2.0 * ((l / (k * k)) / (sin(k) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+259], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * t$95$1), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 18.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 18.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 18.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 18.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 18.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 18.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 18.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 18.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 32.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 32.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 32.3%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 53.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 82.4%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 82.4%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 88.9%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 88.9%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 82.4%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 82.4%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 82.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 82.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 90.4%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 90.4%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 90.4%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 90.3%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 90.3%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 96.7%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 96.8%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 96.8%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   11. Taylor expanded in k around 0 96.8%

       \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

   if 0.0 < (*.f64 l l) < 9.999999999999999e258

   1. Initial program 42.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 42.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 42.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 42.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 44.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 44.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 45.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 45.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 54.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 54.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 54.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 54.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 89.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 89.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 89.1%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 96.0%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 97.4%

         \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot t}} \]

      2. associate-*l/ 97.3%

         \[\leadsto \frac{2}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}}{k \cdot t} \]

   10. Simplified 97.3%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}} \]

   if 9.999999999999999e258 < (*.f64 l l)

   1. Initial program 18.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 18.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 18.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 18.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 18.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 18.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 18.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 18.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 18.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 18.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 18.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 18.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 18.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 58.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 58.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 58.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 60.9%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 60.9%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 58.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 58.0%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 58.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 58.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 67.8%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 67.8%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 67.8%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 67.8%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 67.8%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 75.6%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 75.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 75.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 75.5%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   11. Taylor expanded in l around 0 67.8%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(\sin k \cdot t\right)}\right)} \]
   12. Step-by-step derivation

       1. associate-/r* 75.4%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{2}}}{\sin k \cdot t}}\right) \]

       2. unpow2 75.4%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}}}{\sin k \cdot t}\right) \]

   13. Simplified 75.4%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+259}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right)\\ \end{array} \]

Alternative 3: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 8.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{t_2}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+150}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot 2}{\sin k \cdot t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))) (t_2 (* k (* k t))))
   (if (<= k 8.2e-137)
     (/ (* 2.0 (* l (/ (/ l k) k))) t_2)
     (if (<= k 1.08e+150)
       (* t_1 (* 2.0 (/ (/ l (* k k)) (* (sin k) t))))
       (* t_1 (/ (* l 2.0) (* (sin k) t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double t_2 = k * (k * t);
	double tmp;
	if (k <= 8.2e-137) {
		tmp = (2.0 * (l * ((l / k) / k))) / t_2;
	} else if (k <= 1.08e+150) {
		tmp = t_1 * (2.0 * ((l / (k * k)) / (sin(k) * t)));
	} else {
		tmp = t_1 * ((l * 2.0) / (sin(k) * t_2));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l / tan(k)
    t_2 = k * (k * t)
    if (k <= 8.2d-137) then
        tmp = (2.0d0 * (l * ((l / k) / k))) / t_2
    else if (k <= 1.08d+150) then
        tmp = t_1 * (2.0d0 * ((l / (k * k)) / (sin(k) * t)))
    else
        tmp = t_1 * ((l * 2.0d0) / (sin(k) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double t_2 = k * (k * t);
	double tmp;
	if (k <= 8.2e-137) {
		tmp = (2.0 * (l * ((l / k) / k))) / t_2;
	} else if (k <= 1.08e+150) {
		tmp = t_1 * (2.0 * ((l / (k * k)) / (Math.sin(k) * t)));
	} else {
		tmp = t_1 * ((l * 2.0) / (Math.sin(k) * t_2));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.tan(k)
	t_2 = k * (k * t)
	tmp = 0
	if k <= 8.2e-137:
		tmp = (2.0 * (l * ((l / k) / k))) / t_2
	elif k <= 1.08e+150:
		tmp = t_1 * (2.0 * ((l / (k * k)) / (math.sin(k) * t)))
	else:
		tmp = t_1 * ((l * 2.0) / (math.sin(k) * t_2))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	t_2 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 8.2e-137)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(Float64(l / k) / k))) / t_2);
	elseif (k <= 1.08e+150)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(l / Float64(k * k)) / Float64(sin(k) * t))));
	else
		tmp = Float64(t_1 * Float64(Float64(l * 2.0) / Float64(sin(k) * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	t_2 = k * (k * t);
	tmp = 0.0;
	if (k <= 8.2e-137)
		tmp = (2.0 * (l * ((l / k) / k))) / t_2;
	elseif (k <= 1.08e+150)
		tmp = t_1 * (2.0 * ((l / (k * k)) / (sin(k) * t)));
	else
		tmp = t_1 * ((l * 2.0) / (sin(k) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 8.2e-137], N[(N[(2.0 * N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[k, 1.08e+150], N[(t$95$1 * N[(2.0 * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l * 2.0), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.1999999999999997e-137

   1. Initial program 31.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 31.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 32.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 31.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 32.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 32.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 38.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 38.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 38.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 45.6%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 80.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 80.3%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 67.2%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \]

      2. unpow2 67.2%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

   9. Simplified 67.2%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot k}} \]

   10. Taylor expanded in l around 0 67.2%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. *-rgt-identity 67.2%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} \]

       2. associate-*r/ 67.2%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \]

       3. unpow2 67.2%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{{k}^{2}}\right) \]

       4. unpow2 67.2%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{\color{blue}{k \cdot k}}\right) \]

       5. associate-*l* 74.4%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{1}{k \cdot k}\right)\right)} \]

       6. unpow2 74.4%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{\color{blue}{{k}^{2}}}\right)\right) \]

       7. associate-*r/ 74.4%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot 1}{{k}^{2}}}\right) \]

       8. *-rgt-identity 74.4%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\color{blue}{\ell}}{{k}^{2}}\right) \]

       9. unpow2 74.4%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

   12. Simplified 74.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{k \cdot k}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 74.4%

          \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

       2. associate-*l/ 74.4%

          \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

       3. associate-/r* 77.3%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right)}{k \cdot \left(k \cdot t\right)} \]

   14. Applied egg-rr 77.3%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   if 8.1999999999999997e-137 < k < 1.08e150

   1. Initial program 23.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 23.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 23.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 23.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 25.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 25.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 25.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 25.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 36.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 36.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 36.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 38.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 38.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 79.6%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 79.6%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 79.6%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 79.6%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 79.6%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 79.6%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 79.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 79.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 92.1%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 92.1%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 92.1%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 91.9%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 91.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 91.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 92.0%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 92.0%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 92.0%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   11. Taylor expanded in l around 0 92.0%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(\sin k \cdot t\right)}\right)} \]
   12. Step-by-step derivation

       1. associate-/r* 95.7%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{2}}}{\sin k \cdot t}}\right) \]

       2. unpow2 95.7%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}}}{\sin k \cdot t}\right) \]

   13. Simplified 95.7%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right)} \]

   if 1.08e150 < k

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 34.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 34.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 34.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 34.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 34.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 34.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 34.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 44.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 44.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 44.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 44.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 65.4%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 65.4%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 65.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 65.4%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 75.0%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.0%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 65.4%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 65.4%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 65.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 65.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 69.3%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 69.3%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 69.3%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 69.3%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 69.3%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 85.1%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 85.1%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 85.1%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 85.1%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   11. Step-by-step derivation

       1. frac-times 75.0%

          \[\leadsto \color{blue}{\frac{\ell \cdot \frac{2 \cdot \ell}{\sin k}}{\tan k \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]

   12. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{\frac{\ell \cdot \frac{2 \cdot \ell}{\sin k}}{\tan k \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
   13. Step-by-step derivation

       1. times-frac 85.1%

          \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

       2. associate-/l/ 85.1%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \sin k}} \]

   14. Simplified 85.1%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \sin k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+150}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell \cdot 2}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]

Alternative 4: 79.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.9e-196)
   (* (/ l k) (/ (/ (* l 2.0) (sin k)) (* k (* k t))))
   (* (/ 2.0 k) (/ (/ (* l (/ l (tan k))) (sin k)) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.9e-196) {
		tmp = (l / k) * (((l * 2.0) / sin(k)) / (k * (k * t)));
	} else {
		tmp = (2.0 / k) * (((l * (l / tan(k))) / sin(k)) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.9d-196) then
        tmp = (l / k) * (((l * 2.0d0) / sin(k)) / (k * (k * t)))
    else
        tmp = (2.0d0 / k) * (((l * (l / tan(k))) / sin(k)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.9e-196) {
		tmp = (l / k) * (((l * 2.0) / Math.sin(k)) / (k * (k * t)));
	} else {
		tmp = (2.0 / k) * (((l * (l / Math.tan(k))) / Math.sin(k)) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.9e-196:
		tmp = (l / k) * (((l * 2.0) / math.sin(k)) / (k * (k * t)))
	else:
		tmp = (2.0 / k) * (((l * (l / math.tan(k))) / math.sin(k)) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.9e-196)
		tmp = Float64(Float64(l / k) * Float64(Float64(Float64(l * 2.0) / sin(k)) / Float64(k * Float64(k * t))));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * Float64(l / tan(k))) / sin(k)) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.9e-196)
		tmp = (l / k) * (((l * 2.0) / sin(k)) / (k * (k * t)));
	else
		tmp = (2.0 / k) * (((l * (l / tan(k))) / sin(k)) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.9e-196], N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9000000000000001e-196

   1. Initial program 30.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 31.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 30.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 31.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 31.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 32.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 32.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 41.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 41.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 41.2%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 48.2%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 48.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 80.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 80.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 80.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 86.1%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 86.1%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 80.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 80.2%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 80.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 80.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 85.5%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 85.5%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 85.5%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 85.4%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 85.4%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 91.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 91.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 91.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 91.9%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   11. Taylor expanded in k around 0 79.7%

       \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

   if 1.9000000000000001e-196 < l

   1. Initial program 29.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 29.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 29.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 29.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 29.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 29.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 30.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 30.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 36.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 36.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 36.3%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 38.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 38.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 76.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 76.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 76.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 81.4%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 81.2%

         \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot t}} \]

      2. associate-*l/ 81.2%

         \[\leadsto \frac{2}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}}{k \cdot t} \]

   10. Simplified 81.2%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-196}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}\\ \end{array} \]

Alternative 5: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \tan k} \cdot \left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-99)
   (/ (* 2.0 (* l (/ (/ l k) k))) (* k (* k t)))
   (* (/ l (* k (tan k))) (* (/ (/ 2.0 (sin k)) k) (/ l t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-99) {
		tmp = (2.0 * (l * ((l / k) / k))) / (k * (k * t));
	} else {
		tmp = (l / (k * tan(k))) * (((2.0 / sin(k)) / k) * (l / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.5d-99) then
        tmp = (2.0d0 * (l * ((l / k) / k))) / (k * (k * t))
    else
        tmp = (l / (k * tan(k))) * (((2.0d0 / sin(k)) / k) * (l / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-99) {
		tmp = (2.0 * (l * ((l / k) / k))) / (k * (k * t));
	} else {
		tmp = (l / (k * Math.tan(k))) * (((2.0 / Math.sin(k)) / k) * (l / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.5e-99:
		tmp = (2.0 * (l * ((l / k) / k))) / (k * (k * t))
	else:
		tmp = (l / (k * math.tan(k))) * (((2.0 / math.sin(k)) / k) * (l / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-99)
		tmp = Float64(Float64(2.0 * Float64(l * Float64(Float64(l / k) / k))) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(Float64(l / Float64(k * tan(k))) * Float64(Float64(Float64(2.0 / sin(k)) / k) * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.5e-99)
		tmp = (2.0 * (l * ((l / k) / k))) / (k * (k * t));
	else
		tmp = (l / (k * tan(k))) * (((2.0 / sin(k)) / k) * (l / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.5e-99], N[(N[(2.0 * N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.5000000000000004e-99

   1. Initial program 32.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 32.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 32.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 32.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 33.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 33.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 34.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 34.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 39.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 39.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 39.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 46.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 81.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 81.1%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 67.5%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \]

      2. unpow2 67.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

   9. Simplified 67.5%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot k}} \]

   10. Taylor expanded in l around 0 67.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. *-rgt-identity 67.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} \]

       2. associate-*r/ 67.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \]

       3. unpow2 67.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{{k}^{2}}\right) \]

       4. unpow2 67.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{\color{blue}{k \cdot k}}\right) \]

       5. associate-*l* 75.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{1}{k \cdot k}\right)\right)} \]

       6. unpow2 75.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{\color{blue}{{k}^{2}}}\right)\right) \]

       7. associate-*r/ 75.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot 1}{{k}^{2}}}\right) \]

       8. *-rgt-identity 75.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\color{blue}{\ell}}{{k}^{2}}\right) \]

       9. unpow2 75.5%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

   12. Simplified 75.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{k \cdot k}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 75.5%

          \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

       2. associate-*l/ 75.5%

          \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

       3. associate-/r* 78.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right)}{k \cdot \left(k \cdot t\right)} \]

   14. Applied egg-rr 78.2%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)}} \]

   if 8.5000000000000004e-99 < k

   1. Initial program 24.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 24.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 24.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 24.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 24.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 24.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 24.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 24.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 37.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 37.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 37.4%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 37.4%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 37.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 72.6%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 72.6%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 72.6%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 76.2%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 76.2%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 72.6%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 72.6%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 72.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 72.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 83.5%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 83.5%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 83.5%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 83.5%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 83.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 89.4%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 89.4%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 89.4%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 89.4%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 76.2%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   12. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 65.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

       2. expm1-udef 56.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}\right)} - 1} \]

       3. times-frac 59.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot t}}\right)} - 1 \]

       4. associate-/l* 59.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\color{blue}{\frac{2}{\frac{\sin k}{\ell}}}}{k \cdot t}\right)} - 1 \]

   14. Applied egg-rr 59.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 74.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\right)\right)} \]

       2. expm1-log1p 92.6%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}} \]

       3. associate-/l/ 92.6%

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \tan k}} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t} \]

       4. associate-/r/ 92.6%

          \[\leadsto \frac{\ell}{k \cdot \tan k} \cdot \frac{\color{blue}{\frac{2}{\sin k} \cdot \ell}}{k \cdot t} \]

       5. times-frac 94.9%

          \[\leadsto \frac{\ell}{k \cdot \tan k} \cdot \color{blue}{\left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)} \]

   16. Simplified 94.9%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \tan k} \cdot \left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \tan k} \cdot \left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 6: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+203}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \tan k} \cdot \left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1e+203)
   (* (/ (/ l (tan k)) k) (/ (/ 2.0 (/ (sin k) l)) (* k t)))
   (* (/ l (* k (tan k))) (* (/ (/ 2.0 (sin k)) k) (/ l t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e+203) {
		tmp = ((l / tan(k)) / k) * ((2.0 / (sin(k) / l)) / (k * t));
	} else {
		tmp = (l / (k * tan(k))) * (((2.0 / sin(k)) / k) * (l / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1d+203) then
        tmp = ((l / tan(k)) / k) * ((2.0d0 / (sin(k) / l)) / (k * t))
    else
        tmp = (l / (k * tan(k))) * (((2.0d0 / sin(k)) / k) * (l / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e+203) {
		tmp = ((l / Math.tan(k)) / k) * ((2.0 / (Math.sin(k) / l)) / (k * t));
	} else {
		tmp = (l / (k * Math.tan(k))) * (((2.0 / Math.sin(k)) / k) * (l / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1e+203:
		tmp = ((l / math.tan(k)) / k) * ((2.0 / (math.sin(k) / l)) / (k * t))
	else:
		tmp = (l / (k * math.tan(k))) * (((2.0 / math.sin(k)) / k) * (l / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1e+203)
		tmp = Float64(Float64(Float64(l / tan(k)) / k) * Float64(Float64(2.0 / Float64(sin(k) / l)) / Float64(k * t)));
	else
		tmp = Float64(Float64(l / Float64(k * tan(k))) * Float64(Float64(Float64(2.0 / sin(k)) / k) * Float64(l / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1e+203)
		tmp = ((l / tan(k)) / k) * ((2.0 / (sin(k) / l)) / (k * t));
	else
		tmp = (l / (k * tan(k))) * (((2.0 / sin(k)) / k) * (l / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1e+203], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.9999999999999999e202

   1. Initial program 32.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 32.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 32.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 32.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 33.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 33.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 42.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 42.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.7%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 48.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 81.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 81.1%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 81.2%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 87.1%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 87.1%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 81.1%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 81.1%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 81.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 81.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 85.1%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 85.1%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 85.1%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 85.0%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 85.0%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 92.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 92.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 92.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 92.2%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 87.1%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   12. Applied egg-rr 87.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   13. Step-by-step derivation

       1. times-frac 96.0%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot t}} \]

       2. associate-/l* 95.9%

          \[\leadsto \frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\color{blue}{\frac{2}{\frac{\sin k}{\ell}}}}{k \cdot t} \]

   14. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}} \]

   if 9.9999999999999999e202 < l

   1. Initial program 12.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 12.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 12.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 12.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 12.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 12.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 12.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 12.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 12.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 12.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 12.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 12.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 12.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 58.7%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 58.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 62.1%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 62.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 62.1%

         \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

      2. associate-*r* 58.7%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      3. unpow2 58.7%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      4. associate-/r* 58.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      5. unpow2 58.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

      6. associate-*r* 68.9%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

      7. *-commutative 68.9%

         \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

      8. unpow2 68.9%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

      9. associate-/r* 68.9%

         \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

      10. unpow2 68.9%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

      11. associate-*r* 75.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

      12. associate-*l/ 75.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

      13. associate-*r/ 75.2%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   10. Simplified 75.2%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 62.1%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   12. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 19.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

       2. expm1-udef 19.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}\right)} - 1} \]

       3. times-frac 26.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot t}}\right)} - 1 \]

       4. associate-/l* 26.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\color{blue}{\frac{2}{\frac{\sin k}{\ell}}}}{k \cdot t}\right)} - 1 \]

   14. Applied egg-rr 26.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 26.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\right)\right)} \]

       2. expm1-log1p 78.3%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}} \]

       3. associate-/l/ 78.4%

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \tan k}} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t} \]

       4. associate-/r/ 78.4%

          \[\leadsto \frac{\ell}{k \cdot \tan k} \cdot \frac{\color{blue}{\frac{2}{\sin k} \cdot \ell}}{k \cdot t} \]

       5. times-frac 93.7%

          \[\leadsto \frac{\ell}{k \cdot \tan k} \cdot \color{blue}{\left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)} \]

   16. Simplified 93.7%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \tan k} \cdot \left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+203}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \tan k} \cdot \left(\frac{\frac{2}{\sin k}}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7: 89.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\tan k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l (tan k)) (/ (/ (* l 2.0) (sin k)) (* k (* k t)))))

C

double code(double t, double l, double k) {
	return (l / tan(k)) * (((l * 2.0) / sin(k)) / (k * (k * t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / tan(k)) * (((l * 2.0d0) / sin(k)) / (k * (k * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (l / Math.tan(k)) * (((l * 2.0) / Math.sin(k)) / (k * (k * t)));
}

Python

def code(t, l, k):
	return (l / math.tan(k)) * (((l * 2.0) / math.sin(k)) / (k * (k * t)))

Julia

function code(t, l, k)
	return Float64(Float64(l / tan(k)) * Float64(Float64(Float64(l * 2.0) / sin(k)) / Float64(k * Float64(k * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / tan(k)) * (((l * 2.0) / sin(k)) / (k * (k * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 30.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 30.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 30.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 30.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 31.3%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 39.1%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 43.9%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 43.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 78.4%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 78.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

   2. associate-*l* 84.1%

      \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

8. Applied egg-rr 84.1%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
9. Step-by-step derivation

   1. associate-*l/ 84.1%

      \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   2. associate-*r* 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   3. unpow2 78.4%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   4. associate-/r* 78.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. unpow2 78.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. associate-*r* 83.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   7. *-commutative 83.1%

      \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

   8. unpow2 83.1%

      \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

   9. associate-/r* 83.1%

      \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

   10. unpow2 83.1%

       \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

   11. associate-*r* 90.2%

       \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

   12. associate-*l/ 90.2%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   13. associate-*r/ 90.2%

       \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

10. Simplified 90.2%

    \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

11. Final simplification 90.2%

    \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

Alternative 8: 68.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.75 \cdot 10^{-210}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{k}}{\sin k}}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.75e-210)
   (* 2.0 (/ l (/ (* t (pow k 4.0)) l)))
   (* (/ 2.0 k) (/ (/ (* l (/ l k)) (sin k)) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.75e-210) {
		tmp = 2.0 * (l / ((t * pow(k, 4.0)) / l));
	} else {
		tmp = (2.0 / k) * (((l * (l / k)) / sin(k)) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 2.75d-210) then
        tmp = 2.0d0 * (l / ((t * (k ** 4.0d0)) / l))
    else
        tmp = (2.0d0 / k) * (((l * (l / k)) / sin(k)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.75e-210) {
		tmp = 2.0 * (l / ((t * Math.pow(k, 4.0)) / l));
	} else {
		tmp = (2.0 / k) * (((l * (l / k)) / Math.sin(k)) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 2.75e-210:
		tmp = 2.0 * (l / ((t * math.pow(k, 4.0)) / l))
	else:
		tmp = (2.0 / k) * (((l * (l / k)) / math.sin(k)) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.75e-210)
		tmp = Float64(2.0 * Float64(l / Float64(Float64(t * (k ^ 4.0)) / l)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * Float64(l / k)) / sin(k)) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.75e-210)
		tmp = 2.0 * (l / ((t * (k ^ 4.0)) / l));
	else
		tmp = (2.0 / k) * (((l * (l / k)) / sin(k)) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.75e-210], N[(2.0 * N[(l / N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.75000000000000012e-210

   1. Initial program 31.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 31.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 32.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 31.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 32.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 32.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 42.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 42.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.4%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 48.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in k around 0 61.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. unpow2 61.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 61.0%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   6. Simplified 61.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

   7. Taylor expanded in l around 0 61.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 61.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 61.0%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. associate-/l* 71.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}} \]

   9. Simplified 71.1%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}} \]

   if 2.75000000000000012e-210 < l

   1. Initial program 28.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 28.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 28.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 28.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 28.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 28.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 28.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 28.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 35.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 35.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 35.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 37.6%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 37.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 76.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 76.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 76.0%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

      2. associate-*l* 82.1%

         \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   8. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 81.9%

         \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot t}} \]

      2. associate-*l/ 81.9%

         \[\leadsto \frac{2}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}}{k \cdot t} \]

   10. Simplified 81.9%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}} \]

   11. Taylor expanded in k around 0 71.7%

       \[\leadsto \frac{2}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{k}}}{\sin k}}{k \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.75 \cdot 10^{-210}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{k}}{\sin k}}{k \cdot t}\\ \end{array} \]

Alternative 9: 72.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l k) (/ (/ (* l 2.0) (sin k)) (* k (* k t)))))

C

double code(double t, double l, double k) {
	return (l / k) * (((l * 2.0) / sin(k)) / (k * (k * t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / k) * (((l * 2.0d0) / sin(k)) / (k * (k * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (l / k) * (((l * 2.0) / Math.sin(k)) / (k * (k * t)));
}

Python

def code(t, l, k):
	return (l / k) * (((l * 2.0) / math.sin(k)) / (k * (k * t)))

Julia

function code(t, l, k)
	return Float64(Float64(l / k) * Float64(Float64(Float64(l * 2.0) / sin(k)) / Float64(k * Float64(k * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / k) * (((l * 2.0) / sin(k)) / (k * (k * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 30.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 30.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 30.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 30.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 31.3%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 39.1%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 43.9%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 43.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 78.4%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 78.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]

   2. associate-*l* 84.1%

      \[\leadsto \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

8. Applied egg-rr 84.1%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
9. Step-by-step derivation

   1. associate-*l/ 84.1%

      \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   2. associate-*r* 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   3. unpow2 78.4%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2}} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   4. associate-/r* 78.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. unpow2 78.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. associate-*r* 83.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   7. *-commutative 83.1%

      \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \frac{\ell}{\sin k}\right)} \]

   8. unpow2 83.1%

      \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{\frac{2}{\color{blue}{{k}^{2}}}}{t} \cdot \frac{\ell}{\sin k}\right) \]

   9. associate-/r* 83.1%

      \[\leadsto \frac{\ell}{\tan k} \cdot \left(\color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{\ell}{\sin k}\right) \]

   10. unpow2 83.1%

       \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{\sin k}\right) \]

   11. associate-*r* 90.2%

       \[\leadsto \frac{\ell}{\tan k} \cdot \left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\sin k}\right) \]

   12. associate-*l/ 90.2%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

   13. associate-*r/ 90.2%

       \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

10. Simplified 90.2%

    \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]

11. Taylor expanded in k around 0 75.9%

    \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

12. Final simplification 75.9%

    \[\leadsto \frac{\ell}{k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

Alternative 10: 69.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* k (* k t))) (* l (/ l (* k k)))))

C

double code(double t, double l, double k) {
	return (2.0 / (k * (k * t))) * (l * (l / (k * k)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 / (k * (k * t))) * (l * (l / (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / (k * (k * t))) * (l * (l / (k * k)));
}

Python

def code(t, l, k):
	return (2.0 / (k * (k * t))) * (l * (l / (k * k)))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l * Float64(l / Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / (k * (k * t))) * (l * (l / (k * k)));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 30.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 30.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 30.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 30.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 31.3%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 39.1%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 43.9%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 43.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 78.4%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

7. Taylor expanded in k around 0 66.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 66.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \]

   2. unpow2 66.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

9. Simplified 66.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot k}} \]

10. Taylor expanded in l around 0 66.3%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. *-rgt-identity 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} \]

    2. associate-*r/ 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \]

    3. unpow2 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{{k}^{2}}\right) \]

    4. unpow2 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{\color{blue}{k \cdot k}}\right) \]

    5. associate-*l* 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{1}{k \cdot k}\right)\right)} \]

    6. unpow2 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{\color{blue}{{k}^{2}}}\right)\right) \]

    7. associate-*r/ 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot 1}{{k}^{2}}}\right) \]

    8. *-rgt-identity 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\color{blue}{\ell}}{{k}^{2}}\right) \]

    9. unpow2 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

12. Simplified 71.5%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{k \cdot k}\right)} \]

13. Taylor expanded in k around 0 71.5%

    \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]
14. Step-by-step derivation

    1. unpow2 71.5%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

    2. associate-*r* 71.6%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

15. Simplified 71.6%

    \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

16. Final simplification 71.6%

    \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

Alternative 11: 71.3% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* 2.0 (* l (/ (/ l k) k))) (* k (* k t))))

C

double code(double t, double l, double k) {
	return (2.0 * (l * ((l / k) / k))) / (k * (k * t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 * (l * ((l / k) / k))) / (k * (k * t))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 * (l * ((l / k) / k))) / (k * (k * t));
}

Python

def code(t, l, k):
	return (2.0 * (l * ((l / k) / k))) / (k * (k * t))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 * Float64(l * Float64(Float64(l / k) / k))) / Float64(k * Float64(k * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 * (l * ((l / k) / k))) / (k * (k * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 * N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 30.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 30.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 30.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 30.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 31.3%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 39.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 39.1%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 43.9%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 43.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 78.4%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

7. Taylor expanded in k around 0 66.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 66.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \]

   2. unpow2 66.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

9. Simplified 66.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot k}} \]

10. Taylor expanded in l around 0 66.3%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. *-rgt-identity 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} \]

    2. associate-*r/ 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \]

    3. unpow2 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{{k}^{2}}\right) \]

    4. unpow2 66.3%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{\color{blue}{k \cdot k}}\right) \]

    5. associate-*l* 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{1}{k \cdot k}\right)\right)} \]

    6. unpow2 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{\color{blue}{{k}^{2}}}\right)\right) \]

    7. associate-*r/ 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot 1}{{k}^{2}}}\right) \]

    8. *-rgt-identity 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\color{blue}{\ell}}{{k}^{2}}\right) \]

    9. unpow2 71.5%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

12. Simplified 71.5%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{k \cdot k}\right)} \]
13. Step-by-step derivation

    1. associate-*r* 71.6%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \]

    2. associate-*l/ 71.5%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot \left(k \cdot t\right)}} \]

    3. associate-/r* 73.4%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}\right)}{k \cdot \left(k \cdot t\right)} \]

14. Applied egg-rr 73.4%

    \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)}} \]

15. Final simplification 73.4%

    \[\leadsto \frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{k \cdot \left(k \cdot t\right)} \]

Reproduce

herbie shell --seed 2023240 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))